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**Variational autoencoders are a type of neural network that are often used for unsupervised learning. They are often associated with the autoencoder model because of their architectural affinity, but there are significant differences in the goal and mathematical formulation of the two models.**

The autoencoders are a type of neural network that encodes features and allows for inference, which may be useful for domain-specific tasks. They can be seen as a special case of embedding models. The main difference between the two models is the goal: The autoencoder (also known as hyperparameter learning) aims to learn the parameters of an autoregressive model, while variational autoencoders are trained to encode information using a particle diffusion approach.

A probabilistic variational autoencoder (PAVE) is an example of such a model. A typical PAVE has three hidden units or “eigenvectors”, where each eigenvector encodes a distinct feature, while each input vector has only one eigenvector. An important property of such models is that they can be used to encode arbitrary data without requiring any prior knowledge about the underlying distribution or even uncertainty about its value.****

**The goal of an autoencoder is to learn a representation of a data set in order to be able to reproduce it later. This is done by encoding the data set into a low-dimensional space**

If you’re unfamiliar with the concept of autoencoders, here are a few of their basics. An autoregressive model is one that models the time series data as a function of its previous value, in the form of an autoregressive process. This process is then used to estimate the next value for the series. In contrast, a variational autoencoder (VAE) has been defined as follows:

As you can see, it is an autoregressive model with an additional term that predicts the next value from what has already been observed. Unlike other autoregressive models, this system does not have any initial condition and thus provides more freedom in its prediction.

One advantage is that VAEs have very similar behaviour to linear systems, because they are also made up of linear equations.

So what does it mean for us? Well, there are two main uses for VAEs. First is when we want to approximate some underlying mathematical function where we can’t define our function exactly and need some estimation or approximation methods such as Gaussian quadratic regression where we don’t know exactly where we stand on our function. In these cases we can use these estimates to help us decide how well or poorly we understand our data set.

The second use is when our data set is high dimensional and contains many different kinds of functions and it becomes difficult to compute linear regression on high dimensional data sets such as those produced by deep learning networks (our post here). If this happens then gradient based methods such as Variational Autoencoders may provide the best way to approximate the underlying mathematics which will be easier to interpret than linear regression but will not provide any guarantees about statistical significance since they do not have well defined error bounds like linear regression does.